9711

Lattice of Subgroups of Permutation Groups

One of the simplest and most basic of all algebraic structures is a group. A group is a set with a binary operation that satisfies four axioms: closure, associativity, the existence of an identity, and the existence of inverses. When the operation is commutative, we say that the group is Abelian (in honor of the distinguished Norwegian mathematician Niels Henrik Abel).
Some groups that come to mind contain numbers: the set of integers under addition, the set of nonzero rational numbers under multiplication, or the set under addition mod. Groups occur abundantly in nature as they provide an ideal tool to study symmetry. Any finite group is isomorphic to (structurally the same as) a permutation group.
Many families of groups have been defined in an effort to classify groups. An important insight into the complexities of these families is provided by a diagram of the structure of its subgroups in which one subgroup is joined to another above it if the former is contained in the latter.
This Demonstration shows the lattices of subgroups of four important families of finite groups for selected values of : the symmetric group , of size (order) ; the alternating group , of order ; the dihedral group , of order ; and the cyclic group , of order .
contains all permutations of elements and is generated in Mathematica with the command Permutations[Range[n]] or, in a more compact way, by SymmetricGroup[n] in Version 8. The group is formed by selecting those permutations in that are even, that is, that are obtainable from an even number of interchanges of two elements. The group contains the permutations describing the symmetries of a regular -gon, including the rotations and reflections of the regular -gon and any combination of them. The group is obtained by considering all powers of a single permutation. Cyclic groups are always Abelian. Dihedral groups are non-Abelian for .

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DETAILS

The graph labels do not follow a determined order and are only for the purpose of counting or to help in referring to a particular subgroup. Mouse over a vertex to see a tooltip with the expression of the permutations generating that subgroup.
For example, snapshot 1 shows the lattice of subgroups of , with itself at the top and the identity at the bottom (this last position is common to all lattices). Vertex 6 shows the list , which means that is generated by the two permutations and in cycle notation, or, in Mathematica notation, {Cycle[{{1,2}}], Cycle[{{2,3}}]}. In snapshot 2, we have the lattice of , of order 16 (note that you cannot deduce the order of a group from its lattice of subgroups) formed by its 19 subgroups (including itself and the identity). The subgroup labeled 6 is generated by the permutation Cycle[{{1,3,5,7},{2,4,6,8}}]. In the same lattice, the subgroup labeled 19 is generated by two permutations.
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